Will these DUI in 1980 and 1982 from the pre-computer age still appear on my record when I have a live scan performed?

I cannot give you an answer other than suggest that you contact the CA department of justice for a print out. It is unlikely that these convictions will stop you from getting a job.

Yes they will, which is why you should expunge them. You need to have successfully completed the entire term of probation, have nothing criminal pending, and not be on probation for any other offense(s). Contact a lawyer to file the petition for expungement on your behalf.